#72307622

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18178
commutator, Frattini, cosocle	C₂×#97	C₂⁴
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(3,7)(5,8)(9,10)(11,19,21,32)(12,24)(13,16)(14,29)(15,22,30,35)(17,27)(18,28)(20,31,34,38)(23,37,36,33)(25,39)(26,40);(2,5,6,8)(9,11,12,22,14,23,27,38,13,21,26,35,28,36,39,31)(10,15,17,19,18,20,24,33,16,30,25,32,29,34,40,37);(9,10)(11,20,21,34)(12,25)(13,16)(14,18)(15,23,30,36)(17,26)(19,31,32,38)(22,37,35,33)(24,39)(27,40)(28,29);(1,2)(3,8)(4,6)(5,7)(11,21)(19,32)(23,36)(33,37)

Polycyclic presentation:

2^10|a^2=gh,b^2=f,c^2=gh,e^2=hj,f^2=i,i^2=j,b^a=bf,c^b=cij,d^b=de,d^c=dgh,e^a=eg,e^b=egh,e^c=ej,e^d=ehj,f^a=fi,f^c=fj,f^d=fgj,f^e=fj,g^b=gj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307650, #72307671, #72307751
maximal normal subgroup of	#72307650, #72307671, #72307751

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7*F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);